The theory and simulations of quasi-perpendicular and strictly perpendicular collisionless shocks are reviewed. The text is structured into the following sections and subsections: 1. Setting the frame, where the quasi-perpendicular shock problem is formulated, reflected particle dynamics is described in theoretical terms, foot formation and foot ion acceleration discussed, and the shock potential explained. 2. Shock structure, where the observational evidence is given, and where the simulation studies of quasi-perpendicular shocks are described as far as they deal with shock structure, i.e. the discussion of the different physical shock scales, and their investigation in one-dimensional simulations for small mass-ratios, determination of the shock-transition scale in two ways, from experiment and from simulations which identifies two scales: the foot scale and the ramp scale, the latter being determined from the overshoot magnetic field, and shock reformation is described in one- and two-dimensional simulations, showing that there are regimes when no shock reformation occurs even in supercritical quasi-perpendicular shocks, when the upstream plasma $\beta_i$ is high or when oblique whistlers stabilise the shock in two dimensions; but high Mach number shocks will always become non-stationary. 3. Ion dynamics, describing its role in shock reformation and the various ion-excited instabilities. 4. Electron dynamics, describing electron instabilities in the foot: Buneman and modified two-stream instabilities, generation of electron tails and heating, generation of phase-space holes, and discussion of various wave properties, Weibel instability, 5. The problem of stationarity, posing the theoretical reasons for shocks being non-stationary, discussing nonlinear whistler mediated variability, two-stream and modified two-stream variability, formation of ripples in two-dimensions, 6. Summary and conclusions: The possibility of shock breaking.